Stochastically forced zonal flows

This thesis investigates the dynamics of multiple zonal jets, that spontaneously emerge on the barotropic beta-plane, driven by a homogenous and rapidly decorrelating forcing and damped by bottom drag. Decomposing the barotropic vorticity equation into the zonal-mean and eddy equations, and neglecting the eddy-eddy interactions, defines the quasi-linear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system. Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is a homogenous eddy field with no jets. When the forcing is also isotropic in space, we characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is up to a factor of six smaller than that for QL zonostrophic instability.;The constraint of isotropic forcing is then relaxed and spatially anisotropic forcing is used to drive the jets. Meridionally drifting jets are observed whenever the forcing breaks an additional symmetry that we refer to as mirror, or reflexional symmetry. The magnitude of drift speed in our results shows a strong variation with both mu and beta: while the drift speed decreases almost linearly with decreasing mu, it actually increases as beta decreases. Similar drifting jets are also observed in QL, with the same direction (i.e. northward or southward) and similar magnitude as NL jet-drift. Starting from the laminar solution, and assuming a mean-flow that varies slowly with reference to the scale of the eddies, we obtain an approximate equation for the vorticity correlation function that is then solved perturbatively. The Reynolds stress of the pertubative solution can then be expressed as a function of the mean-flow and its y-derivatives. In particular, it is shown that as long as the forcing breaks mirror-symmetry, the Reynolds stress has a wave-like term, as a result of which the mean-flow is governed by a dispersive wave equation.;In a separate study, Reynolds stress induced by an anisotropically forced unbounded Couette flow with uniform shear gamma, on a beta-plane, is calculated in conjunction with the eddy diffusivity of a co-evolving passive tracer. The flow is damped by linear drag on a time scale mu--1. The stochastic forcing is controlled by a parameter alpha, that characterizes whether eddies are elongated along the zonal direction (alpha 0) or are isotropic (alpha = 0). The Reynolds stress varies linearly with alpha and non-linearly and non-monotonically with gamma; but the Reynolds stress is independent of beta. For positive values of alpha, the Reynolds stress displays an "anti-frictional" effect (energy is transferred from the eddies to the mean flow) and a frictional effect for negative values of alpha. With gamma = beta =0, the meridional tracer eddy diffusivity is v'2/(2mu), where v' is the meridional eddy velocity. In general, beta and gamma suppress the diffusivity below v'2/(2mu).